What major reason would Britain, France, and Russia have in allying in the Triple Entente?

They feared the increasing economic and military might of the German Empire which had achieved unification in 1871, after the defeat of France in the Franco-Prussian War.

As a late comer to the colonial distribution of the world, which threatened British and French interests in Africa and Asia, or even Russian interests, Germany wanted a fairer share of the colonialist feast. Its growing economy and industrial power allowed it to have a larger and more powerful military. It had ambitions of more influence; it was an emerging power whose ambitions and interests clashed with those of the hegemon of the system, Great Britain.

Another reason for this coalition was the traditional diplomatic and geopolitical concern of many European powers of balancing against any single power that could become too strong and dominant on the continent; this is particularly true of Great Britain, the holder of the key of the balance of power in Europe for many centuries.